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Enumeration of Intersection Graphs of x-Monotone Curves

Authors Jacob Fox, János Pach, Andrew Suk

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LIPIcs.GD.2024.4.pdf
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Subject Classification

ACM Subject Classification
  • Mathematics of computing → Combinatorics
Keywords
  • Enumeration
  • intersection graphs
  • pseudo-segments
  • x-monotone

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Abstract

A curve in the plane is x-monotone if every vertical line intersects it at most once. A family of curves are called pseudo-segments if every pair of them have at most one point in common. We construct 2^Ω(n^{4/3}) families, each consisting of n labelled x-monotone pseudo-segments such that their intersection graphs are different. On the other hand, we show that the number of such intersection graphs is at most 2^O(n^{3/2-ε}), where ε > 0 is a suitable constant. Our proof uses an upper bound on the number of set systems of size m on a ground set of size n, with VC-dimension at most d. Much better upper bounds are obtained if we only count bipartite intersection graphs, or, in general, intersection graphs with bounded chromatic number.

Cite As Get BibTex

Jacob Fox, János Pach, and Andrew Suk. Enumeration of Intersection Graphs of x-Monotone Curves. In 32nd International Symposium on Graph Drawing and Network Visualization (GD 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 320, pp. 4:1-4:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.GD.2024.4

Author Details

Jacob Fox
  • Department of Mathematics, Stanford University, CA, USA
János Pach
  • HUN-REN Alfréd Rényi Institute of Mathematics, Budapest, Hungary
Andrew Suk
  • Department of Mathematics, University of California San Diego, La Jolla, CA, USA

Funding

  • Fox, Jacob: supported by NSF Award DMS-2154129.
  • Pach, János: Rényi Institute, Budapest. Supported by NKFIH grant K-176529, Austrian Science Fund Z 342-N31 and ERC Advanced Grant "GeoScape."
  • Suk, Andrew: Supported by NSF CAREER award DMS-1800746, and NSF awards DMS-1952786 and DMS-2246847.

Acknowledgements

We would like to thank Zixiang Xu for pointing out the reference [N. Alon et al., 2017] to us.

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